1

large cooling tower (Figure 27.17) are shown in Figures 27.18 to 27.24 for some of the important loading conditions discussed in Section 27.5. The finite element model used considers the shell to be fixed at the top of the columns and, thus, does not account for the effect of the concentrated column reactions. Also, in considering the analyses under the individual loading conditions, it should be remembered that the effects are to be factored and combined to produce design values.

The dead load analysis results in Figure 27.18 and Figure 27.19 indicate that the shell is always under compression in both directions, except for a small circumferential tension near the top. This is a very desirable feature of this geometrical form and the result of a complex shape finding process [6,11].

In Figures 27.20 to 27.22, the results of an analysis for a quasistatic wind load using the K1.0 distribution from Figure 27.11 are shown. Large tensions in both the meridional and circumferential directions are present. The regions of tension may extend a considerable distance along the circumference from the windward meridian, and the magnitude of the forces is strongly dependent on the distribution selected. In contrast to bluff bodies, where the magnitude of the extensional force along the meridian would be essentially a function of the overturning moment, the cylindrical-type body is also strongly influenced by the circumferential distribution of the applied pressure, a function of the surface roughness. The major effect of the shearing forces is at the level of the lintel where they are transferred into the columns. The internal suction effects, Figure 27.23 and Figure 27.24, are significant only in the circumferential direction.

For the service temperature case shown in Figure 27.25 and Figure 27.26, the main effects are bending in the lower region of the shell wall.

27.6.3 Stability

The analysis of hyperbolic cooling towers for instability or buckling is a subject that has been investigated for several decades [12]. Shell buckling is a complex topic to treat analytically in any case, due to the influence of imperfections; for reinforced concrete, it is even more difficult. While the governing

0 400 kN/m FIGURE 27.18 Circumferential forces n11D under deadweight.

0 600 kN/m

FIGURE 27.19 Meridional forces n22D under deadweight.

0 600 kN/m

FIGURE 27.19 Meridional forces n22D under deadweight.

FIGURE 27.20 Circumferential forces n11W under wind load.

0 400 kN/m

FIGURE 27.21 Meridional forces n22W under wind load.

0 400 kN/m

FIGURE 27.21 Meridional forces n22W under wind load.

FIGURE 27.22 Shear forces n12W under wind load.

FIGURE 27.23 Circumferential forces n11S under internal suction.

equations may be generalized to treat instability by using nonlinear strain-displacement relations and thereby introducing the geometric stiffness matrix, the correlation between the resulting analytical solutions and the possible failure of a reinforced concrete cooling tower is questionable.

Nevertheless, it has been common to analyze cooling tower shells for instability under an unfactored combination of dead load plus wind load plus internal suction. The corresponding instability buckling pattern is shown in Figure 27.27.

Interaction diagrams calibrated from experimental studies based on bifurcation buckling are also available [4,5,13]. Additionally, there are empirical methods based on wind tunnel tests that consider a snap-through buckle at the upper edge at each stage of construction [4]. These formulas are proportional to h/R and are convenient for establishing an appropriate shell thickness profile. If buckling safety is evaluated based on such a linear buckling analysis or an experimental investigation, the buckling safety